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The design and manufacturing of the first model of an International Linear Collider (ILC) Main Linac superconducting quadrupole is in progress at Fermilab. The quadrupole has a 78 mm aperture, a 36 T integrated gradient, and a cold mass length of 700 mm. A superferric magnet configuration with iron poles and four racetrack coils was chosen based on magnet performance, cost, and reliability considerations. Each coil is wound using enamel insulated, 0.5 mm diameter, NbTi superconductor. The quadrupole package also includes shell type dipole steering coils. The results of the quadrupole design, including magnetic and mechanical analyses, are presented. Specific issues related to the quadrupole magnetic center stability, superconductor magnetization and mechanical stability are discussed and analyzed. The magnet quench protection system, current leads, and mounting the quadrupole inside ILC Main Linac cryomodule will also be briefly discussed.